Coefficients -- which roughly is a measure of the degree of relatedness/shared genes between a given dog and its ancestors -- are determined by applying a mathematical formula called Wright's Coefficient of Inbreeding (COI).

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It is the probability that a homozygous gene pair will be identical by descent from both sides of a pedigree. In the formula, FX is your dog's COI, FA is that of the ancestor common to both sides of the pedigree. n1 and n2 are the numbers of generations on each side between your dog and that ancestor. In other words, if your dog Flux is a double-grandson of FAbulous it tells you how likely it is you can get exactly the same gene passed down to Flux through each of his parents.

Here's more from where that came from, a primer on breeding coefficients: